Multilevel subdivision techniques are presented for the efficient
numerical approximation of complicated dynamical behavior. Concretely
we develop adaptive methods which allow to extract statistical
information on the underlying dynamical system. This is done by
an approximation of natural invariant measures as well as (almost)
cyclic dynamical components. We discuss issues concerning the
implementation (e.g. parallelization strategies) and indicate
potential applications of these methods (e.g. to the computation
of Lyapunov exponents). The results are illustrated by several
numerical examples.